Neutral zone classifiers include 'no-decision' as a classification outcome. This paper extends neutral zone classifiers to sequential contexts for analyzing longitudinal data. Applications could include medical diagnosis where a decision variable is repeatedly measured on each subject with the expectation of being able to ultimately identify a patient disease status. Sequential classifiers monitor the sequence of measurements and decide when to stop sampling and how to classify the subject. Bayesian sequential classification rules make classifications on the basis of the minimum expected loss. This approach is a challenge due to computational complexity associated with evaluating the expected future costs. We consider Gaussian contexts. In the homogeneous case we demonstrate the equivalence between sequential Bayesian classifier and a simpler boundary-based framework. A solution for the heterogeneous Gaussian case is presented using the boundary-based framework. A recursive algorithm is developed to efficiently determine decision boundaries that minimize the overall expected cost.

Alternative decision boundaries which are competitive with the optimal decision boundaries are studied. Misclassification rates and expected sample size are investigated and the results are compared with non-sequential classifiers.

Cumulative sum (CUSUM) algorithms are used for monitoring in various applications, including manufacturing, network monitoring, financial markets, biosurveillance, and many more. A popular CUSUM technique for detecting a change in the in-control distribution of an independent

data sequence is based on repeated use of the sequential probability ratio test (SPRT). Some optimality results have been derived for the SPRT-based CUSUM when the in-control and out-of-control distributions are fully known. We introduce an approximation formula for the

threshold value of an SPRT-based CUSUM. Limited research has been performed on CUSUM techniques when the distributions are not fully specified. This research is concerned about how to use the CUSUM when the underlying in-control distribution is arbitrary and unknown, and the out-of-control density is either an additive or a multiplicative transformation of the in-control density. The proposed solution combines an adaptive nonparametric kernel density estimator derived from an in-control sample of observations with a smoothed bootstrap algorithm that enables the CUSUM to work effectively for reasonably sized sets of in-control data.

The context of comparing two different groups of subjects that are measured repeatedly over time is considered. Our specific focus is on highly variable count data which have a non-negligible frequency of zeros and have time trends that are difficult to characterize. These challenges are often present when analyzing bacteria or gene expression data sets. Traditional longitudinal data analysis methods, including Generalized Estimating Equations, can be challenged by the features present in these types of data sets. We propose a Bayesian methodology that effectively confronts these challenges. A key feature of the methodology is the use of Gaussian Processes to flexibly model the time trends. Inference procedures based on both sharp and interval null hypotheses are discussed, including for the important hypotheses that test for group differences at individual time points. The proposed methodology is illustrated with next generation sequencing data sets corresponding to two different experimental conditions.

Statistical process control (SPC) methods are widely used in many fields, such as manufacturing, engineering and scientific studies aiming to reduce both financial and time costs through detecting abnormal process changes and making real-time decisions. Hence, the performance of a SPC method is usually measured by its detection time, and we favor the method that can detect a deterioration in the process as early as possible. In this dissertation, we propose SPC methods for monitoring two commonly encountered types of data, namely, Type I censored Weibull lifetime data and autocorrelated count data. Both methods are shown having significantly shorter detection time than existing methods.

In the first half of this dissertation, we first propose a SPC method for monitoring the Weibull scale parameter in the context of Type I censored data, assuming the shape parameter is fixed. The proposed method is a Shewhart-type method based on a likelihood ratio test (LRT) that utilizes an exponentially weighted moving average of the log-likelihood. In previous literature, this type of method is called a weighted exponentially weighted moving average (WEWMA) method. The WEWMA method is compared with a more standard exponentially weighted moving average (EWMA) method and a cumulative sum (CUSUM) method which were studied as alternative solutions to the monitoring problem. Numerical results show that the WEWMA method often performs better than these two alternatives. Sensitivity studies show that the WEWMA method is more robust to variations in batch sizes and censoring times. Moreover, two good properties of the WEWMA method are proved. A rust-resistant example is used to illustrate the proposed WEWMA method. We then extend the WEWMA and the CUSUM methods to the context where the shape parameter also needs to be monitored. Joint charts that simultaneously look for a change in either parameter are proposed.

In the second half of this dissertation, we deal with a problem in monitoring autocorrelated count data. Poisson integer valued autoregressive (INAR) models have been showed as an effective method to model autocorrelated count data without overdispersion and Poisson lognormal (PLN) INAR models extend their use to overdispersed contexts. We propose the use of a repeated sequential probability ratio test (SPRT) procedure to detect a change in first-order INAR and PLN INAR models. We consider a change in the mean, the autocorrelation parameter, and the overdispersion parameter. Simulation results show the repeated SPRT procedure performs favorably relative to previously proposed CUSUM procedures that are based on either the observations themselves or residuals of the observations from predicted values. A data set on invasive insect species is used to illustrate the repeated SPRT procedure.

Group sequential clinical trials offer administrative, economic, and ethical benefits over fixed sample methods when testing a treatment versus control. Traditional methods based on the assumption that the treatment distribution is a pure shift of the control distribution may not always hold. The possibility that an individual from the treatment group may not respond to the treatment motivates the use of a mixture distribution for the treatment group. This work considers two test procedures based on the Wilcoxon Rank Sum statistic for a group sequential design to detect the one-sided mixture alternative. Error spending functions are used for the allocation of error rates at each stage. The two tests are evaluated individually in determination of critical values and arm sizes and asymptotic multivariate normality is shown to hold for both. Upon comparison, the tests are presented to be asymptotically equivalent. Both test statistics maintain the Type I error rate even if F is misspecified in the design alternative. A more general definition of the treatment effect is used with the mixture distribution. Various estimators for the treatment effect are evaluated.

Network surveillance algorithms are becoming increasingly important as the ability to monitor a wide variety of data is rapidly expanding. Traffic metrics are usually count data that display a non-stationary pattern in their mean structure. We propose to model traffic counts using a generalized linear mixed model to capture these features. We then develop three tracking statistics proposed for anomaly detection. Two of the statistics are derived variants of a generalized likelihood ratio approach, which itself is not computationally tractable. The first of these variants is based on an approximation to the integrated likelihood while the second is based on the concept of h-likelihood. We also consider a tracking statistic that is an exponentially weighted moving average. We investigate the properties of the three tracking statistics from the point of view of false alarm rate and detection power, and compare the proposed tracking statistics with current literature. Our comparisons show that the two generalized likelihood ratio variants are preferred choices as SPC tools for network surveillance. Computational aspects of the three procedures are also discussed.

Three well known methods for constructing prediction intervals in a generalized linear mixed model (GLMM) are the methods based on pseudo-likelihood, Laplace, and Quadrature approximations. All three of these methods are available in the SAS procedure GLIMMIX. We propose a new method based on a mean squared error (MSE) approximation of the empirical best predictor. Following the approach by Harville and Kackar (1984) for a linear mixed model (LMM), we decompose the prediction error into two terms for the purpose of deriving the MSE approximation. Unlike in the LMM case, however, closed form expressions for the two terms in the subsequent MSE approximation are not available. We confront the computational challenge by proposing a Monte Carlo algorithm for evaluating the plug-in estimators of these two terms. For three illustrated examples, the negative binomial, the Poisson and the Bernoulli GLMMs, numerical results show our prediction interval methodology improves the coverage probability over the three methods available in GLIMMIX. Moreover, our results show that with bootstrap adjustments, our method achieves coverage probabilities satisfactorily close to the nominal level.

Group sequential designs play an important role in monitoring clinical trials. In this dissertation, we consider the construction of one-sided group sequential designs where the stopping rule includes boundaries for early stopping to accept for futility and to reject for efficacy. The traditional assumption that all patients have the same likelihood of benefiting from the treatment is sometimes unrealistic and it can underestimate the required sample size. This motivates us to power the design for an alternative where the treatment group observations come from a mixture of normal distributions. For the proposed setting, we use standardized test statistics based on sample means. Stopping boundaries and arm size for the design are determined by Type I and Type II error spending equations. The unknown variance case is discussed, and we introduce the group sequential t-test for the mixture setting. With the mixture model, we discuss a more general definition of treatment effect. The maximum likelihood estimator and method of moment estimator for the treatment effect are discussed.

The cluster analysis has been widely applied to many fields. In this dissertation, Hot spot detection, as an important application of the spatial clustering, is thoroughly introduced and the current methodologies used in hot spot detection are presented and compared. In addition, we introduce a model based scan method to identify hot spots on spatial lattice arrays. Four features introduced by the proposed methodology are: (1) A Generalized Linear Mixed Model (GLMM) that provides a realistic model for correlated count data; (2) A border comparison that is used to determine the significance of a candidate hot spot at each stage of sequential searches; (3) A confirmation step better separates the homogeneous and heterogeneous hot spots; (4) An iterative process that finds secondary hot spots by conditioning on previously found hot spots. A heuristic search algorithm (MBSM-H) is proposed that reduces the high computational demands associated with a global search algorithm (MBSM-G). Both algorithms are illustrated through simulated examples and an application to Integrated Pest Management where an orchard is assessed for potential pest problems. Comparisons between the model based scan method and alternative methodologies are presented in this dissertation as well.

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